The data obtained from an auto-correlation measurement is known to be valuable for witnessing various forms of non-classicality, including the negativity of the P-function and quantum non-Gaussianity [1]}, [2]}.
We now show that the knowledge of \(\mathbf {p}\)  can reveal the negativity of the Wigner function [3]}, arguably the strongest form of non-classically for a bosonic mode. In particular, the negativity of the Wigner function implies the negativity of the P-function [4]}.
Similarly, it implies that the corresponding state is non-Gaussian, as a Gaussian state has a Gaussian (and thus positive) Wigner functionIn addition, Hudson's theorem [5]} tell us that any pure state with a positive Wigner function is Gaussian.. Thus, demonstrating Wigner negativity for a light source brings evidence of its strong quantum nature. Note that it has been shown recently that witnesses of Wigner negativity can be derived systematically using a hierarchy of semi definite programs [6]}. Our contribution is more specific and aims at witnessing Wigner negativity simply and directly from \(\hat{P}_1(\mathbf {p})\) .
